Spatiotemporal patterns of neural fields in a spherical cortex with general connectivity

The human brain consists of billions of neurons and these neurons pool together
in groups at different scales. On one hand, these neural entities tend to behave
as single units and on the other hand show collective macroscopic patterns of activity.
The neural units communicate with each other and process information over time.
This communication is through small electrical impulses which at the macroscopic
scale are measurable as brain waves. The electric field that is produced collectively
by macroscopic groups of neurons within the brain can be measured on the surface
of the skull via a brain imaging modality called Electroencephalography (EEG). The
brain as a neural system has variant connection topology, in which an area might not
only be connected to its adjacent neighbors homogeneously but also distant areas can
directly transfer brain activity [16]. Timing of these brain activity communications
between different neural units bring up overall emerging spatiotemporal patterns.
The dynamics of these patterns and formation of neural activities in cortical surface
is influenced by the presence of long-range connections between heterogeneous neural
units. Brain activity at large-scale is thought to be involved in the information processing
and the implementation of cognitive functions of the brain. This research
aims to determine how the spatiotemporal pattern formation phenomena in the brain
depend on its connection topology. This connection topology consists of homogeneous
connections in local cortical areas alongside the couplings between distant functional
units as heterogeneous connections. Homogeneous connectivity or synaptic weight
distribution representing the large-scale anatomy of cortex is assumed to depend on
the Euclidean distance between interacting neural units. Altering characteristics of
inhomogeneous pathways as control parameters guide the brain pattern formation
through phase transitions at critical points. In this research, linear stability analysis
is applied to a macroscopic neural field in a one-dimensional circular and a twodimensional
spherical model of the brain in order to find destabilization mechanism
and subsequently emerging patterns.